\( z_1, z_2, z_3 \) represent the vertices A, B, C of a triangle ABC respectively in the Argand plane. If
\[ |z_1 - z_2| = \sqrt{25 - 12 \sqrt{3}}, \] \[ \left|\frac{z_1 - z_3}{z_2 - z_3}\right| = \frac{3}{4}, \] \[ \text{and } \angle ACB = 30^\circ, \]
Then the area (in sq. units) of that triangle is:
\( z_1, z_2, z_3 \) represent the vertices A, B, C of a triangle ABC respectively in the Argand plane. If
\[ |z_1 - z_2| = \sqrt{25 - 12 \sqrt{3}}, \] \[ \left|\frac{z_1 - z_3}{z_2 - z_3}\right| = \frac{3}{4}, \] \[ \text{and } \angle ACB = 30^\circ, \]
Then the area (in sq. units) of that triangle is:
Show Hint
- $\frac{3}{2}$
- $3$
- $5$
- $\frac{5}{2}$
The Correct Option is B
Solution and Explanation
Top Questions on Complex numbers
Let \( z \) satisfy \( |z| = 1, \ z = 1 - \overline{z} \text{ and } \operatorname{Im}(z)>0 \)
Then consider:
Statement-I: \( z \) is a real number
Statement-II: Principal argument of \( z \) is \( \dfrac{\pi}{3} \)
Then:
- AP EAPCET - 2025
- Mathematics
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If \( z \) and \( \omega \) are two non-zero complex numbers such that \( |z\omega| = 1 \) and
\[ \arg(z) - \arg(\omega) = \frac{\pi}{2}, \]
Then the value of \( \overline{z\omega} \) is:
- AP EAPCET - 2025
- Mathematics
- Complex numbers
If \( x^a y^b = e^m, \)
and
\[ x^c y^d = e^n, \]
and
\[ \Delta_1 = \begin{vmatrix} m & b \\ n & d \\ \end{vmatrix}, \quad \Delta_2 = \begin{vmatrix} a & m \\ c & n \\ \end{vmatrix}, \quad \Delta_3 = \begin{vmatrix} a & b \\ c & d \\ \end{vmatrix} \]
Then the values of \( x \) and \( y \) respectively (where \( e \) is the base of the natural logarithm) are:
- AP EAPCET - 2025
- Mathematics
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- If \( \alpha \) is the common root of the quadratic equations \( x^2 - 5x + 4a = 0 \) and \( x^2 - 2ax - 8 = 0 \), where \( a \in \mathbb{R} \), then the value of \( \alpha^4 - \alpha^3 + 68 \) is:
- AP EAPCET - 2025
- Mathematics
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- If \( \omega_1 \) and \( \omega_2 \) are two non-zero complex numbers and \( a, b \) are non-zero real numbers such that \[ |a\omega_1 + b\omega_2| = |a\omega_1 - b\omega_2|, \] then \( \dfrac{\omega_1}{\omega_2} \) is:
- AP EAPCET - 2025
- Mathematics
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Questions Asked in AP EAPCET exam
- If the binding energy per nucleon of deuteron (\( ^1\mathrm{H}^2 \)) is 1.15 MeV and an \(\alpha\)-particle has a binding energy of 7.1 MeV per nucleon, then the energy released per nucleon in the given reaction is \[ ^1\mathrm{H}^2 + ^1\mathrm{H}^2 \rightarrow ^2\mathrm{He}^4 + Q \]
- AP EAPCET - 2025
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- If the function \( f \) defined by \[ f(x) = \begin{cases} \dfrac{1 - \cos 4x}{x^2}, & x<0 \\ a, & x = 0 \\ \dfrac{\sqrt{x}}{\sqrt{16 + \sqrt{x}} - 4}, & x>0 \end{cases} \] is continuous at \( x = 0 \), then \( a = \)
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If \( \vec{u}, \vec{v}, \vec{w} \) are non-coplanar vectors and \( p, q \) are real numbers, then the equality:
\[ [3\vec{u} \quad p\vec{v} \quad p\vec{w}] - [p\vec{v} \quad \vec{w} \quad q\vec{u}] - [2\vec{w} \quad q\vec{v} \quad q\vec{u}] = 0 \]
holds for:
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Statement-I: In the interval \( [0, 2\pi] \), the number of common solutions of the equations
\[ 2\sin^2\theta - \cos 2\theta = 0 \]
and
\[ 2\cos^2\theta - 3\sin\theta = 0 \]
is two.
Statement-II: The number of solutions of
\[ 2\cos^2\theta - 3\sin\theta = 0 \]
in \( [0, \pi] \) is two.
- AP EAPCET - 2025
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If \( A \) and \( B \) are acute angles satisfying
\[ 3\cos^2 A + 2\cos^2 B = 4 \]
and
\[ \frac{3 \sin A}{\sin B} = \frac{2 \cos B}{\cos A}, \]
Then \( A + 2B = \ ? \)
- AP EAPCET - 2025
- Trigonometric Identities
